Complexity of virtual multistrings

A virtual [Formula: see text]-string [Formula: see text] consists of a closed, oriented surface [Formula: see text] and a collection [Formula: see text] of [Formula: see text] oriented, closed curves immersed in [Formula: see text]. We consider virtual [Formula: see text]-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual [Formula: see text]-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [Formula: see text]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual [Formula: see text]-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [Formula: see text]-strings in general, but Gibson found a counterexample for [Formula: see text]-strings. We show that Kadokami’s statement holds for non-parallel [Formula: see text]-strings and exhibit a counterexample for general virtual [Formula: see text]-strings.